Question

What is the slope and the y -intercept of the equation 9+27 over 2x = −3 over 2y. Write the y -intercept as an ordered pair

The slope is ________. The y-intercept is (__)

Answer 1

clearly, the user meant
9 + 27/(2x) = −3/(2y)
this simplifies to
9/x + 1/y + 6 = 0
which is a hyperbola, and has no x- or y-intercepts

so the question was poorly posed.

Answer 2

To find the slope and the y-intercept of the equation \( \frac{9 + 27}{2x} = -\frac{3}{2}y \), let's first simplify and rearrange the equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. Simplify the left side:
\[
\frac{9 + 27}{2x} = \frac{36}{2x} = \frac{18}{x}
\]
Thus, the equation becomes:
\[
\frac{18}{x} = -\frac{3}{2}y
\]

2. Now, clear the fraction by multiplying both sides by \( -2x \):
\[
-2 \times 18 = -3y \times x
\]
\[
-36 = -3xy
\]
Or equivalently:
\[
3xy = 36
\]

3. Solve for \( y \):
\[
y = \frac{36}{3x} = \frac{12}{x}
\]

Next, we want to write this in a more familiar form; to express it in terms of \( y \):
\[
y = \frac{12}{x}
\]

This equation doesn’t quite fit the \( y = mx + b \) form because it's a hyperbola rather than a linear equation. Therefore it does not have a traditional slope or y-intercept as defined in linear equations.

Since he's looking for the slope and y-intercept in that context, we can note:

- The equation does not yield a linear relationship. Thus, there is no slope or y-intercept in the conventional sense.

To summarize:
- The slope is undefined.
- There is no y-intercept as it does not represent a linear function.

**Final Answer:**
- The slope is undefined. The y-intercept does not exist in the linear context.

Answer 3

Thank you for the clarification! Let's go through the equation step by step with your correction:

Given the equation:
\[
9 + \frac{27}{2x} = -\frac{3}{2y}
\]

We can rearrange it as follows:

1. First, let's bring all terms to one side:
\[
\frac{27}{2x} + \frac{3}{2y} + 9 = 0
\]

2. To simplify this, we can multiply through by \(2xy\) (to eliminate the denominators), which gives us:
\[
27y + 3x + 18xy = 0
\]

3. Rearranging this, we have:
\[
18xy + 27y + 3x = 0
\]

4. Now we can express \(y\) in terms of \(x\):
\[
y(18x + 27) = -3x
\]
\[
y = \frac{-3x}{18x + 27}
\]

This is indeed a hyperbola, and it does not yield a linear equation, so there is no traditional slope or y-intercept.

**Conclusion:**
- The equation can be interpreted as a hyperbola, which means it does not have x- or y-intercepts in the linear sense.
- The question was indeed poorly posed if we are strictly looking for slope and y-intercept, as those apply to linear equations.

If you have any more specific questions or need further assistance, feel free to ask!